Consider an urn holding N balls which are either white or black. Whenever I take out a ball it is replaced randomly with either a white or black ball,
with fixed probability p for the replacement ball being white.
I am playing this game for quite a while (so that the start configuration no longer matters) and now I am taking out n white balls in a row (n < N).
What is the probability that the next ball is also white?
If p is (near) 1/2 then one could make these two arguments:
1) If we take out a sequence of n white balls it indicates that there are probably
more white balls in the urn than black balls (due to a random fluctuation in the replacement process), so the next ball is most likely also white: P(white) > P(black).
2) If we take out n white balls, the ratio of white to black necessarily decreases,
so it is more likely that the next is actually black: P(black) > P(white).
What do you think? And does it make a difference if we actually know p and N ?
added later: I have posted the solution now as a comment, but I have to warn you that this is fun only if you really try to find an answer yourself first.